On the \({\mathcal {A}}\)-equivalence of quasi-ordinary parameterizations

Abstract

We study the analytic equivalence of quasi-ordinary hypersurfaces in \({\mathbb {C}}^{r+1}\) by means of its normalized quasi-ordinary parameterization. In this context, two quasi-ordinary hypersurfaces are analytic equivalent if and only if their normalized quasi-ordinary parameterizations are \({\mathcal {A}}\)-equivalent. We introduce the set \(\Lambda _{H}^{\mathcal {D}}\subset {\mathbb {N}}^{r}\) associated to Kähler r-forms that generalizes an important analytic invariant of plane branches and allows us to identify terms in a normalized quasi-ordinary parameterization that can be eliminable by an element of \({\mathcal {A}}\)-group.

Keywords

Quasi-ordinary hypersurface Kähler r-forms Jacobian ideal

Mathematics Subject Classification

Notes

Acknowledgements

We thank the referee for the observations made that helped us to improve the redaction of this paper and mainly for pointing out an important correction in Definition 2.3. The first author was partially supported by CNPq and the second one by CAPES.